The problem comes from Alan Pollack's Differential Topology, pg. 5. Suppose that X is a k-dimensional manifold. Show that every point in X has a neighborhood diffeomorphic to all of $\Bbb{R}^k$.

I have already shown that $\Bbb{R}^k$ is diffeomorphic to $B_a$ (part (a) of the question) the open ball of radius $a$, though have little to no understanding of how to proceed.

Thank you

The author defines a k-manifold as a set such that each point possesses a neighborhood diffeomorphic to an open set of $\Bbb{R}^k$

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    $\begingroup$ What is his definition of a manifold? For many people, having neigbourhoods diffeomorphic to $\mathbb R^k$ is the definition. $\endgroup$ – Fredrik Meyer Sep 2 '13 at 14:07
  • $\begingroup$ Sorry, I added his definition to the question. $\endgroup$ – Pax Kivimae Sep 2 '13 at 14:11

You already did all the work. If $x\in X$, then there is a neighborhood $U$ of $x$ that is diffeomorphic to an open set of $\mathbb{R}^k$. Take $\phi: U\to V$ to be the chart. Since $V$ is open, there is an open ball $B_a$ centered at $x$ contained in $V$. Take $W=\phi^{-1}(B_a)$. Then $W$ is diffeomorphic to $B_a$, and as you already proved, this is diffeomorphic to $\mathbb{R}^k$.

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    $\begingroup$ Ah, thank you very much. It's amazing how the simple solutions can elude you. $\endgroup$ – Pax Kivimae Sep 2 '13 at 14:19
  • $\begingroup$ Dear Pax: No problem :) $\endgroup$ – rfauffar Sep 2 '13 at 14:20
  • $\begingroup$ One interesting observation is that if you ask for holomorphic coordinate functions and change $\mathbb{R}^k$ for $\mathbb{C}^k$ and diffeomorphic to holomorphic (aka, a complex manifold), then the result is no longer true! $\endgroup$ – rfauffar Sep 2 '13 at 14:21
  • $\begingroup$ Huh, weird. Guess I should check out a few more books... $\endgroup$ – Pax Kivimae Sep 2 '13 at 14:30

Given a point $p \in \mathbb{R}^k$, and an open neighbourhood $U$ of $p$, there exists an open set $V$ with $p \in V \subset U$ such that $V$ is diffeomorphic to the $k$-dimensional unit ball. Just take the Euclidean metric on $\mathbb{R}^k$, and let $V$ be a sufficiently small open ball centred on $p$.

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